In this talk, Paul Raff goes over five years of mathematics research in a quick 48 minutes to cover all that he has accomplished during graduate school. The talk is split into three parts. The first part of the talk concerns the problem of avoiding prescribed differences. The second part of the talk concerns the problem of finding the number of spanning trees in arbitrary grid graphs. The final part of the talk concerns the firefighter problem, a dynamical graph theory problem that tries to model the spread of fires. This talk took place on August 14, 2009 at 11am in room 705 of the Hill Center at Rutgers University, New Brunswick. The thesis advisor is Doron Zeilberger, and Fred Roberts, Vladimir Retakh, and Neil Sloane were on the committee.